A note on certain classes of transformation
formulas involving several variables
R.K. Raina
R.K. Ladha
Abstract
This paper gives certain new classes of transformation formulas in the
form of multiple-series identities involving several variables. The results obtained, besides being capable of unifying and providing extensions to various
transformation and reduction formulas, also yield other new formulas. The
applicability of the main results is treated briefly in the concluding section.
1 Introduction
Transformation and reduction (or summation) formulas relating to special functions
of one or several variables are of utmost importance. These are invariably used in
different applied branches of theoretical physics and engineering sciences. See, for
example [1], [3], [7], [8], and [9].
While working on an alternative proof of a recently posed problem concerning a certain hypergoemetric identity, Grosjean and Srivastava [2] were lead to its
multiple-series generalization and to its other related extensions. Several transformation and reduction formulas of hypergeometric functions have been deduced in [2]
from the main multiple-series identities. Multiple-series identities have also recently
been obtained in [5], [8], and were also recorded in [9]. Our motive in this paper
is to obtain new classes of transformation formulas in the form of certain general
Received by the editors February 1996.
Communicated by A. Bultheel.
1991 Mathematics Subject Classification : 33 C 20, 33 C 35.
Key words and phrases : Reduction formulas, hypergeometric series, multiple-series, Lauricella’s functions, Gauss summation theorem, Hermite polynomials, classical orthogonal polynomials.
Bull. Belg. Math. Soc. 4 (1997), 517–524
518
R.K. Raina – R.K. Ladha
multiple-series identities (Theorems 1-4, below). These results can be applied to
substantially more general classes of special functions and orthogonal polynomials.
Applications are discussed briefly in the concluding section of this paper.
Denoting by (ap ) the p-dimensional vector of complex numbers (a1, . . . , ap ), and
let the Pochhammer symbol (λ)n stand for
(1.1)
Γ(λ + n)
(λ)n =
=
Γ(λ)
(
1,
λ(λ + 1) . . . (λ + n − 1),
if n = 0,
if n ∈ {1, 2 . . .}
)
,
then the generalized hypergeometric series in r variables z1, . . . , zr is defined by ([9,
p. 38, eqn. (24)])


p:p1;...;pr
(1.2) Fq:q

1 ;...;qr

(ap ) : (b1p1 ); . . . ; (brpr );

z1 , . . . , zr 
(cq ) : (d1q1 ); . . . ; (drqr );
∞
X
=
f(m1 , . . . , mr )
m1 ,...,mr =0
r
Y
{zimi /mi !},
i=1
where
p
Y
(1.3)
f(m1 , . . . , mr ) =
(ai )m1 +...mr
i=1
q
Y
(ci )m1 +...+mr
 pk
Y



(bk )m

r
Y  i=1 i k






qk
Y


k

 (di )mk 


k=1 

i=1
.
i=1
When r = 1, (1.2) reduces to the generalized hypergeometric function of one variable, and for r = 2, it reduces to the generalized Kampé de Fériet function of two
variables.
For further details about these functions, see [9, Chapter 1]. In our sequel we
shall also be using the triple hypergeometric series of Srivastava (see [9,p. 44])
F (3)[x, y, z] which provides unification of Lauricella’s fourteen hypergeometric functions F1, F2 , . . . , F14 and additional functions HA , HB and HC .
2 Main results
For non-negative integers mj and qj we define
[mi /qi]
(2.1)
Ai (mi , qi) =
X
(−mi )qi ki Bi (mi , ki ),
(i ∈ {1, . . . , r}),
ki =0
where [x] denotes the greatest integer in x, and Bi (mi , ki )(mi , ki ≥ 0,
i ∈ {1, . . . , r}) are bounded sequences of real (or complex) parameters.
Consider the multiple-series
(2.2)
I=
∞
X
m1 ,...,mr =0
∆(m1, . . . , mr )
r
Y
i=1
(
)
xmi
Ai (mi, qi ) i
,
mi !
A note on certain classes of transformation formulas involving several variables 519
where ∆(m1, . . . , mr ) is a bounded multiple sequence of real (or complex) parameters, and |xi | ≤ Ri (Ri > 0, ∀i ∈ {1, . . . , r}),
Using (2.1), and the elementary relation
(2.3)
(−m)k =

(−1)k m!/(m − k)!,
0 ≤ k ≤ m,
k > m,
0,
then (2.2) gives
(2.4)
∞
X
I=
∆(m1, . . . , mr ) ·
m1 ,...,mr =0

r [mX
i /qi ]
Y
i=1

ki =0


(−1)qi ki
i
Bi (mi , ki )xm
i .
(mi − qi ki )!
The absolute convergence of the series involved permits change in the order of summation, and with this assumption making an appeal to the series arrangement property [10, Lemma 3, p. 10], we are lead to the following multiple-series identity :
Theorem 1. Corresponding to the bounded sequence Bi (mi , ki ), let Ai(mi , qi ) be
defined by (2.1), ∀i ∈ {1, . . . , r), where mi , qi are non-negative integers. Then
(2.5)
∞
X
∆(m1, . . . , mr )
m1 ,...,mr =0
r
Y
(
i=1
∞
X
=
xmi
Ai (mi , qi) i
mi !
)
∆(m1 + q1k1 , . . . , mr + qr kr )
m1 ,k1 ,...,mr ,kr =0
·
r
Y
i=1
(
i +qi ki
xm
i
qi ki
(−1) Bi (mi + qi ki , ki )
mi !
)
,
where, ∆(m1, . . . , mr ) is a single-valued, bounded multiple sequence of real (or complex) parameters such that each of the series involved is absolutely convergent.
Following [2], we consider other interesting variations of Theorem 1. To this end,
let us put the sequence
(2.6)
∆(m1, . . . , mr ) = Ω(m1, . . . , mr )
r
Y
{mi !},
i=1
and
(2.7)
Bi (mi , ki ) = ci (ki )/mi !,
so that from (2.1), we write
(2.8)
A∗i (mi, qi ) =
[mi /qi]
X
(−mi )qi ki Ci (ki )/mi !,
ki =0
then Theorem 1 yields the following :
(∀i ∈ {1, . . . , r}),
520
R.K. Raina – R.K. Ladha
Theorem 2. For non-negative integers mi , qi(i = 1, . . . , r) there exists the identity :
∞
X
(2.9)
Ω(m1 , . . . , mr )
m1 ,...,mr =0
i
{A∗i (mi , qi )xm
i }
i=1
∞
X
=
r
Y
Ω(m1 + q1 k1 , . . . , mr + qr kr ) ·
r
Y
(
(−1)
qi ki
i=1
m1 ,k1 ,...,mr ,kr =0
xmi +qi ki
Ci (ki ) i
mi !
)
,
where, A∗i (mi, qi ) is defined by (2.8), and Ω(m1 , . . . , mr ) is a bounded multiple sequence of real (or complex) parameters such that each of the series is absolutely
convergent.
If we set the sequence
Ω(m1 , . . . , mr ) = C ∗(m1 + . . . + mr ),
(2.10)
and invoke the elementary identity
(2.11)
∞
X
f(m1 + . . . + mr )
m1 ,...,mr =0
r
Y
{zimi /mi !} =
∞
X
(z1 + . . . + zr )m
m=0
i=1
f(m)
,
m!
then Theorem 2 gives the result :
Theorem 3. Under the assumptions of Theorem 2, there exists the identity
(2.12)
∞
X
C ∗(m1 + . . . + mr )
m1 ,...,mr =0
r
Y
i
{A∗i (mi, qi )xm
i }
i=1
=
∞
X
C ∗(m + q1k1 + . . . + qr kr )
k1 ,...,kr ,m=0
·
r
(x1 + . . . + xr )m Y
{Ci (ki )(−xi )qi ki },
m!
i=1
where C ∗(m) is an arbitrary bounded sequence such that the series involved in (2.12)
converge absolutely.
Another variation of Theorem 2 can be contemplated if we put
(2.13)
Ω(m1, . . . , mr ) = C ∗(m1 + . . . + mr )(γ1 )m1 . . . (γr )mr ,
and
xi = x (i = 1, . . . , r),
(2.14)
in Theorem 2, and make use of the identity [10, p. 61, eqn. (9)] :
∞
X
(2.15)
m1 ,...,mr =0
f(m1 + . . . + mr )
r
Y
i=1
=
i
{(γ1 )mi xm
i /mi !}
∞
X
m=0
f(m)(γ1 + . . . + γr )m
xm
,
m!
to simplify the multiple m-series on the right side of (2.9), and this way we are lead
to the following :
A note on certain classes of transformation formulas involving several variables 521
Theorem 4. Under the assumptions of Theorem 2, there exists the identity
∞
X
∗
C (m1 + . . . + mr )
m1 ,...,mr =0
=
r
Y
{A∗i (mi , qi )(γi )mi xmi }
i=1
∞
X
∗
C (m + q1 k1 + . . . + qr kr )
k1 ,...,kr ,m=0
·(γ1 + q1k1 + . . . + γr + qr kr )m
r
Y
xm
m!
{(γr )qi ki Ci (ki )(−xi)qi ki },
i=1
provided that the series involved in (2.16) converge absolutely.
3 Applications
On account of the presence of arbitrary sequences, our Theorems 1-4 would widely be
applicable and thus lead to numerous interesting transformation and reduction (or
summation) formulas. We shall consider some cases of deductions, thereby revealing
the usefulness of our main results.
We observe that if
qi = 2 (i = 1, . . . , r),
and
(3.1)
Ci (ki ) =
2−2ki
,
ki !(λi + 1/2)ki
then from (2.8) by Gauss summation theorem [10, p.30, eqn. (7)], we have
(3.2)
A∗i (mi , qi ) =
2mi (λi )mi
,
mi !(2λi )mi
then Theorems 2, 3 and 4 give the recent known results of Grosjean and Srivasta [2,
pp. 289-291].
It is worth noticing that most of the reduction formulas recorded in SrivastaKarlsson’s monograph [9, Chapter 1] would emerge from our Theorems 1 and 2
by suitably specializing the arbitrary sequences in accordance with the involvement
of various summation theorems relating to the hypergeometric functions listed, for
instance, in [4, Chapter 7]. To illustrate, we deduce one such reduction formula from
Theorem 1.
For qi = 1, we put
(3.3)
Bi (mi , ki ) =
(ai )ki (bi )ki
,
(ci )ki (1 + ai + bi − ci − mi )ki ki !
so that from [4, p. 539, entry 88] :
(3.4)
Ai(mi , 1) =
(ci − ai )mi (ci − bi )mi
.
(ci )mi (ci − ai − bi )mi
522
R.K. Raina – R.K. Ladha
Theorem 1 gives then
(3.5)
∞
X
∆(m1, . . . , mr )
m1 ,...,mr =0
r
i
Y
(ci − ai )mi (ci − bi )mi xm
i
{
i=1
∞
X
=
(ci )mi (ci − ai − bi )mi mi!
}
∆(m1 + k1 , . . . , mr + kr )
m1 ,k1 ,...,mr ,kr =0
r
Y
·
{
i=1
(−1)ki (ai)ki (bi )ki
xmi +ki }.
(ci )ki (1 + ai + bi − ci − mi − ki )ki mi !ki ! i
If r = 1, and
(3.6)
∆(m) =
(α1 )m . . . (αP )m
,
(β1 )m . . . (βQ)m
then (3.5) in conjunction with the definition (1.2) assumes the form


P
:1;2
(3.7) FQ+1
:0;1 
: c − a − b ; a, b ;
(αP )
(βQ), c − a − b : − − −
; c


= P +2 FQ+2 

x, x 

;
(αP ), c − a, c − b;
(βQ), c, c − a − b;


x ,P ≤ Q
Next, for qi = 2, we set,
Ci (ki ) = (−1/4t2i )ki /ki !,
(3.8)
in (2.8) to get
A∗i (mi, 2) = (2ti )mi Hmi (ti)/mi !,
(3.9)
where Hn (t) denotes the Hermite polynomials [10,p.40]. With the substitutions (3.8)
and (3.9), Theorem 3 yields
(3.10)
∞
X
∗
C (m1 + . . . + mr )
m1 ,...,mr =0
r
Y
{(xi/2ti )mi Hmi (ti )/mi !}
i=1
=
∞
X
C ∗ (m + 2k1 + . . . + 2kr )
k1 ,...,kr ,m=0
r
(x1 + . . . + xr )m Y
·
{(−x2i /4t2i )ki /ki !}.
m!
i=1
When r = 2, (3.10) takes the form
∞
X
(3.11)
m,n=0
(x/2u)m (y/2v)n
m!
n!
∞
m
X
(x + y) (−x2/4u2 )n (−y 2/4v 2 )p
=
.
A(m + 2n + 2p)
m!
n!
p!
m,n,p=0
A(m + n)Hm (u)Hn (v)
A note on certain classes of transformation formulas involving several variables 523
Lastly, when qi = 1, and
(3.12)
Ci (ki ) =
(λi )ki
,
(µi )ki ki !
in (2.8) then from Gauss summation theorem [10,p.30, eqn. (9)], we have
A∗i (mi , 1) =
(3.13)
(µi − λi )mi
,
(µi )mi mi !
and choosing the sequence C ∗(m) same as in (3.6), we arrive at the following result
from Theorem 4 :


(αP ) : γ1 , µ1 − λ1 ; . . . , γr , µr − λr ;

P :2;...;2
(3.14) FQ:1;...;1


x, . . . , x 
(βQ) :
µ1 ; . . . ; µr
;
(α1 )m . . . (αp )m
xm
=
(γ1 + . . . + γr )m
m!
m=0 (β1 )m . . . (βQ )m
∞
X


P +1:2;...;2
· FQ+1:1;...;1

P
(αP + m) ,
γi + m : γ1 , λ1 ; . . . , γr , λr ;
P
(βQ + m) ,
γi
:
µ1 ; . . . ; µr

−x, . . . , −x 
,
;
P
where P ≤ Q, and γi = γ1 + . . . + γr .
If P = Q = 0, and r = 2, then in view of [9,p.28, eqn.(31)], (3.14) in terms of
the triple hypergeometric series gives

(3.15)


α, γ − β ;

x 
 2 F1 
2 F1 
γ


= F (3) 

λ, v − µ ;
;
v
α + λ :: −;
−
−

x 

;
; −; −; α, β ; λ, µ ;
:: −; α + λ ; −; −;
γ
;
v

x, −x, −x 

;
We conclude this paper by remarking that numerous other transformation and reduction (or summation) formulas involving various special functions can be deduced
from Theorems 1-4. More importantly, as pointed out in the derivation of (3.11), the
specialisations of arbitrary sequences in Theorems 1-4 can be set as in [6], yielding
summation formulas for different classical orthogonal polynomials also.
524
R.K. Raina – R.K. Ladha
References
[1] Exton H., Multiple Hypergeometric Functions and Applications, Halsted Press
(Ellis Horwood, Chichester), John Wiley and Sons, New York, London, Sydney and Toronto, 1976.
[2] Grosjean C. and Srivastava H.M., Some transformation and reduction formulas for hypergeometric series in several variables, J. Comput. Appld. Math.
37 (1991), 287-299.
[3] Niukkanen A.W., Generalized hypergeometric series NF (x1, . . . , xN ) arising
in physical and quantum chemical applications, J. Phys. A : Math. Gen. 16
(1983), 1813-1825.
[4] Prudnikov A.P., Brychkov Yu.A. and Marichev O.I., Integrals and Series,
Volume 3 : More special Functions, Gordon & Breach, New York 18, 1990.
[5] Raina R.K., On certain combinatorial multiple-series identities, Comp. Rend.
Bulg. Acad. Sci. 41 (3) (1988), 13-15.
[6] Srivastava H.M., The Weyl fractional integral of a general class of polynomials,
Boll. Un. Mat. Ital. (6) 2-B (1983), 219-228.
[7] Srivastava H.M., Reduction and summation formulae for certain classes of
generalized hypergeometric series arising in physical quantum chemical applications, J. Phys. A : Math. Gen. 18 (1985), 3079-3085.
[8] Srivastava H.M., A class of hypergeometric polynomials, J. Austral. Math. Soc.
43 (Ser. A) (1987), 187-198.
[9] Srivastava H.M. and Karlsson P.W., Multiple Gaussian Hypergeometric Series,
Halsted Press (Ellis Horwood, Chichester, Brisbane and Toronto, 1985.
[10] Srivastava H.M. and Manocha H.L., A Treatise on Generating Functions, Halsted Press (Ellis Horwood, Chichester) , John Wiley & Sons, New York, Chichester, Brisbane and Toronto,1984.
R.K. Raina
Department of Mathematics C.T.A.E., Campus
Udaipur - 313 001, Rajasthan, INDIA
R.K. Ladha
Department of Mathematics, Govt. College,
Sirohi - 307001, Rajasthan, INDIA